Monsieur Charles Swann is artistically inclined (but primarily as a collector), musically gifted (though sharpest as a critic), and “a particular friend of the Comte de Paris”. The appearance of a painting from his collection (on loan at the Corot) in the pamphlet for the Figaro serves—en fin de compte—as nothing more than an occasion for his abasement at the hands of the narrator’s jealous great-aunt. His artistic talents are squandered on the decoration of old society ladies’ drawing rooms. In his occasional spare moments, he tinkers with an ever-unfinished essay on Vermeer of Delft.
Odette de Crécy, on the other hand, arouses in him—at least at first—nothing more than feelings of indifference.
It’s no wonder, then, that what finally moves Swann’s heart—what sets in motion a helpless, protracted infatuation—is Swann’s sudden recognition, in Odette, of a likeness to a figure with ancient significance: Zipporah, Jethro’s daughter, as she appears in Botticelli’s The Youth of Moses.
Swann’s newfound attachment to Odette quickly becomes a source of torment, as Odette’s reciprocation of his love falters. Where once Odette proclaimed, with delight, that “You know, you will never be like other people!”, she now sighs, “Ah! so you never will be like other people!”; where she once wondered, “I do wish I could find out what there is in that head of yours!”, she now exclaims in frustration, “Oh, I do wish I could change you; put some sense into that head of yours.”
The Verdurin family, once “far more intelligent, far more artistic, surely, than the people one knows” (in virtue of their having facilitated Swann and Odette’s courtship) become “the most perfect specimens of their disgusting class” (after forsaking Swann, and introducing Odette to a rival). While once Odette would reassure him, nightly, that “We shall meet, anyhow, to-morrow evening; there’s a supper-party at the Verdurins’,” she now pleads, “We sha’n’t be able to meet to-morrow evening; there’s a supper-party at the Verdurins’.”
Swann snoops outside Odette’s lighted window, captivated by its “mysterious golden juice” hours after she’d sent him away for the night pleading fatigue. He invokes the hospitality of an old friend with a country house, just to be near Odette on a weekend trip to which he was not invited, ultimately baffling his host as he spends each evening “inspect[ing] the dining-rooms of all the hotels in Compiègne”. Swann declines the Baron de Charlus’ offer of accompaniment to the Marquise de Saint-Euverte’s glitzy party, entreating the Baron, instead, to check on Odette as “she goes to see her old dressmaker”.
As his jealousy reaches a breaking point, Swann, in one telling scene, scrutinizes a sealed envelope with the responsibility of whose delivery Odette has entrusted him:
He lighted a candle, and held up close to its flame the envelope which he had not dared to open. At first he could distinguish nothing, but the envelope was thin, and by pressing it down on to the stiff card which it enclosed he was able, through the transparent paper, to read the concluding words… He took a firm hold of the card, which was sliding to and fro, the envelope being too large for it and then, by moving it with his finger and thumb, brought one line after another beneath the part of the envelope where the paper was not doubled, through which alone it was possible to read.
On this occasion, as on many, Swann is simply distraught: “he took off his spectacles, wiped the glasses, passed his hands over his eyes.”
Swann’s pursuit of Odette occupies the largest chapter of Swann’s Way, the first among the seven volumes comprising Marcel Proust’s enormous In Search of Lost Time. Though much of Swann’s Way chronicles the rich emotional memories of its narrator, this chapter, alone, drags, at great pain, through the details of Swann’s trials.
Perhaps a profound message is to be extracted from this trying chapter.
The following problem, communicated to me by a friend, appears to be part of the internet-math folklore.
Player 1 writes down any two distinct numbers on separate clips of paper. Player 2 randomly chooses one of these slips of paper and looks at the number. Player 2 must decide whether the number in his hand is the larger of the two numbers.
Game. You are playing a game with an adversary, consisting of the following steps. The adversary chooses two unequal numbers, writes them down on separate pieces of paper, and then places these into separate sealed envelopes. The adversary then flips a fair coin to determine which of the two envelopes to give you. After seeing the number in your envelope, you must decide whether the hidden number is larger than or smaller than the number in your hand. You win if you can do this with probability strictly higher than .
Here is the winning strategy:
Strategy. Choose any function which takes values between 0 and 1, and which is forever increasing in value. (Mathematically, this means some function which is strictly order-preserving in the sense that whenever . For example, the “logistic” function satisfies this property.)
Now call the number you’re shown x. State that the hidden number is smaller than x larger probability f(x). Otherwise, state that it is larger.
Claim. The Strategy satisfies the requirement put forth in the Game.
Proof: Supposing that the adversary has already chosen his two numbers, written them down, and placed them into the envelopes, let’s call these two numbers s and l, for smaller and larger. Breaking down the likelihood of winning along the two possible outcomes of the coin flip, we see that:
where in the final step we used the increasingness of f. ♦
Here’s where things get interesting. Consider the bogus claim below (an asterisk denotes that a claim or proof is not valid).
Claim*: The Strategy does not meet the requirement of the Game.
Proof*: Without loss of generality, let’s call the number you’ve been shown x. Breaking down the likelihood of winning as in the proof of the Claim, we have that:
Yet the Game demands a likelihood of winning strictly above . ∗
There are various ways of explaining what exactly went wrong in this bogus proof. Perhaps the most transparent is to point out that the symbol x here is being made to stand for either among two possible values, here—the higher and the lower of the adversary’s two numbers—depending on where exactly in the proof it turns up.
A better way of putting it is to point out that “the number you’ve been shown” hasn’t yet been defined at the time of the phrase’s utterance, or that, more accurately, it refers not yet to a number but rather to a random variable which can take either of two possible values with equal likelihood. (This is the same fallacy as that behind the related “Two envelopes paradox“.)
This game illuminates Swann’s quandary. Even in the face of complete uncertainty, it demonstrates, an advantage over random chance can be achieved, through a strategy which, after the resolution of the uncertain outcome, reacts accordingly.
On the other hand, by naming and making static the unknown variable before the resolution of the uncertainty, one precipitates an apparently unwinnable struggle against randomness. This move, of course, is faulty, and obscures that the facts of the matter change accordingly as the uncertain outcome is resolved.
Swann insists on understanding Odette as a fixed, static entity. In reality, he faces two: one who “would kiss him before the eyes of his coachman” and another who “in a towering rage, broke a vase”. In attempting to resolve an uncertainty—that of Odette’s heart—before its resolution’s appointed time, Swann recasts a surmountable game as doomed.
The solution, of course, is not to see, by candlelight, into the adversary’s envelopes. It’s to react properly once your envelope is revealed.
* * *
Early on in Swann’s Way, the reader is made aware of Swann’s having made “a most unsuitable marriage”, to a woman known to “dye her hair and redden her lips”. Even as the novel reaches back into Swann’s past, however, the identity of the future Mme. Swann—in other words, whether she’s in fact Odette de Crécy—is artfully concealed. This uncertainty is ultimately resolved.
Throughout the entire account of Swann’s love for Odette, meanwhile, one force remains constant: the magical role played by “the andante movement of Vinteuil’s sonata for the piano and violin” in his feelings. Having once heard the piece many years prior, and yet unaware of its name, when Swann, astounded, hears it once again at the Verdurins’, it quickly becomes “the national anthem of their love”.
Swann at one point utters a peculiar thought: that of being “agonised by the reflection, at the moment when [the piece] passed by him, so near and yet so infinitely remote, that, while it was addressed to their ears, it knew them not…”. I too, have lamented that, no matter how deeply I may come to know mathematics’ abstract terrain, it will never be aware of my presence.
Much later, however, Swann reverses course. “For he had no longer, as of old,” the narrator remarks, “the impression that Odette and he were not known to the little phrase. Had it not often been the witness of their joys?”
Does not this particular bit of mathematics—this random, and yet winnable, game—suggest likewise that, to the contrary, math can know one’s presence? Though random chance may yet unfold, it leaves small clues in the process, small differences around us, features which would be altered had chance unfolded differently. By responding wisely to these subtle differences, we unwittingly interact with the great mechanisms of chance, and gain an imperceptible advantage. Perhaps they too detect our presence in turn.